odule in parallel lines and trianglesimages.pcmac.org/sisfiles/schools/al/bessemercity/... · two...

? ESSENTIAL QUESTION

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Angle Relationships in Parallel Lines and Triangles

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Math On the Spot

How can you use angle relationships in parallel lines and triangles to solve real-world problems?

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Many cities are designed on a grid with parallel streets. If another street runs across the parallel lines, it is a transversal. Special relationships exist between parallel lines and transversals.

21MODULE

LESSON 21.1

Parallel Lines Cut by a Transversal

8.EE.7b, 8.G.5

LESSON 21.2

Angle Theorems for Triangles

8.EE.7, 8.EE.7b,

8.G.5

LESSON 21.3

Angle-Angle Similarity

8.EE.6, 8.EE.7,

8.G.5

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Complete these exercises to review skills you will need

for this module.

Solve Two-Step Equations

EXAMPLE 7x + 9 = 30

7x + 9 - 9 = 30- 9

7x = 21

7x __ 7

= 21 __

7

x = 3

Solve for x.

1. 6x + 10 = 46

2. 7x - 6 = 36

3. 3x + 26 = 59

4. 2x + 5 = -25

5. 6x - 7 = 41

6. 1 _ 2

x + 9 = 30

7. 1 _ 3

x - 7 = 15

8. 0.5x - 0.6 = 8.4

Name AnglesEXAMPLE

Give two names for the angle formed by the dashed rays.

9. 10. 11.

Use three points of an angle, including the vertex, to name the angle. Write the vertex between the other two points: ∠JKL or ∠LKJ.You can also use just the vertex letter to name the angle if there is no danger of confusing the angle with another. This is also ∠K.

Write the equation.

Subtract 9 from both sides.

Simplify.

Divide both sides by 7.

Simplify.

Unit 10664

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Reading Start-Up

Active ReadingPyramid Before beginning the module,

create a pyramid to help you organize what

you learn. Label each side with one of the

lesson titles from this module. As you study

each lesson, write important ideas like

vocabulary, properties, and formulas

on the appropriate side.

VocabularyReview Words

✔ acute angle (ángulo agudo)

✔ angle (ángulo) congruent (congruente)✔ obtuse angle (ángulo

obtuso) parallel lines (líneas

paralelas)✔ vertex (vértice)

Preview Words alternate exterior angles

(ángulos alternos externos)

alternate interior angles (ángulos alternos internos)

corresponding angles (ángulos correspondientes (para líneas))

exterior angle (ángulo externo de un polígono)

interior angle (ángulos internos)

remote interior angle (ángulo interno remoto)

same-side interior angles (ángulos internos del mismo lado)

similar (semejantes) transversal (transversal)

Visualize VocabularyUse the ✔ words to complete the graphic. You can put more

than one word in each section of the triangle.

Understand VocabularyComplete the sentences using preview words.

1. A line that intersects two or more lines is a .

2. Figures with the same shape but not necessarily the same size

are .

3. An is an angle formed by one side of

the triangle and the extension of an adjacent side.

Reviewing Angles

B in

∠ABC

measures > 0°

and < 90°

measures >90° and < 180°

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1 23 4

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74°

70°

36° 36°

70°

3

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Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module.

What It Means to YouYou will learn about the special angle relationships formed when

parallel lines are intersected by a third line called a transversal.

What It Means to YouYou will use the angle-angle criterion to determine similarity

of two triangles.

Which angles formed by the

transversal and the parallel lines

seem to be congruent?

It appears that the angles below

are congruent.

∠1 ≌ ∠4 ≌ ∠5 ≌ ∠8

∠2 ≌ ∠3 ≌ ∠6 ≌ ∠7

Explain whether the triangles are similar.

Two angles in the large triangle are congruent to two angles in the

smaller triangle, so the third pair of angles must also be congruent,

which makes the triangles similar.

70° + 36° + m∠3 = 180°

m∠3 = 74°

Angle Relationships in Parallel Lines and Triangles

GETTING READY FOR

Use informal arguments to

establish facts about the angle

sum and exterior angle of

triangles, about the angles

created when parallel lines

are cut by a transversal, and

the angle-angle criterion for

similarity of triangles.

Key Vocabularytransversal (transversal)

A line that intersects two or

more lines.

Use informal arguments to

establish facts about the angle

sum and exterior angle of

triangles, about the angles

created when parallel lines

are cut by a transversal, and

the angle-angle criterion for

similarity of triangles.

EXAMPLE 8.G.5

EXAMPLE 8.G.5

8.G.5

8.G.5

Visit my.hrw.com to see all CA Common Core Standards unpacked.

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EXPLORE ACTIVITY 1

ESSENTIAL QUESTION

Parallel Lines and TransversalsA transversal is a line that intersects two lines in the same plane at two

different points. Transversal t and lines a and b form eight angles.

What can you conclude about the angles formed by parallel lines that are cut by a transversal?

L ESSON

21.1Parallel Lines Cut by a Transversal

Use geometry software to explore the angles formed

when a transversal intersects parallel lines.

Construct a line and label two points on the line

A and B.

Create point C not on ‹

__

› AB . Then construct a line

parallel to ‹

__

› AB through point C. Create another

point on this line and label it D.

A

B

8.G.5

8.G.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Also 8.EE.7b

Angle Pairs Formed by a Transversal

Term Example

Corresponding angles lie on the same side of the transversal t, on the same side of lines a and b. ∠1 and ∠5

Alternate interior angles are nonadjacent angles that lie on opposite sides of the transversal t, between lines a and b.

∠3 and ∠6

Alternate exterior angles lie on opposite sides of the transversal t, outside lines a and b. ∠1 and ∠8

Same-side interior angles lie on the same side of the transversal t, between lines a and b. ∠3 and ∠5

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EXPLORE ACTIVITY 1 (cont’d)

File

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Edit Display Construct Transform Measure Graph Window Help

E

F

H

G

Create two points outside the two parallel

lines and label them E and F. Construct

transversal ‹

__

› EF . Label the points of

intersection G and H.

Measure the angles formed by the parallel

lines and the transversal. Write the angle

measures in the table below.

Drag point E or point F to a different

position. Record the new angle measures

in the table.

Angle ∠CGE ∠DGE ∠CGH ∠DGH ∠AHG ∠BHG ∠AHF ∠BHF

Measure

Measure

ReflectMake a Conjecture Identify the pairs of angles in the diagram. Then

make a conjecture about their angle measures. Drag a point in the

diagram to confirm your conjecture.

1. corresponding angles

2. alternate interior angles

3. alternate exterior angles

4. same-side interior angles

C

D

E

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EXPLORE ACTIVITY 2

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Justifying Angle RelationshipsYou can use tracing paper to informally

justify your conclusions from the first

Explore Activity.

Lines a and b are parallel. (The black

arrows on the diagram indicate

parallel lines.)

Trace the diagram onto tracing paper.

Position the tracing paper over the original diagram so that ∠1 on

the tracing is over ∠5 on the original diagram. Compare the two

angles. Do they appear to be congruent?

Use the tracing paper to compare all eight angles in the

diagram to each other. List all of the congruent angle pairs.

A

B

C

Finding Unknown Angle MeasuresYou can find any unknown angle measure when two parallel lines are cut by

a transversal if you are given at least one other angle measure.

Find m∠2 when m∠7 = 125°.

∠2 is congruent to ∠7 because they

are alternate exterior angles.

Therefore, m∠2 = 125°.

Find m∠VWZ.

∠VWZ is supplementary to

∠YVW because they are

same-side interior angles.

m∠VWZ + m∠YVW = 180°

EXAMPLEXAMPLE 1

A

B

How does a decrease in the measure of ∠1

affect the other angle measures?

Math TalkMathematical Practices

8.G.5

8.EE.7b

Recall that vertical angles are the opposite angles formed by two intersecting lines. ∠1 and ∠4 are vertical angles.

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From the previous page, m∠VWZ + m∠YVW = 180°, m∠VWZ = 3x°, and

m∠YVW = 6x°.

m∠VWZ + m∠YVW = 180°

3x° + 6x° = 180°

9x = 180

9x ___ 9

= 180 ____ 9

x = 20

m∠VWZ = 3x° = (3 · 20)° = 60°

Guided Practice

Use the figure for Exercises 1–4. (Explore Activity 1 and Example 1)

1. ∠UVY and are a pair of corresponding angles.

2. ∠WVY and ∠VWT are angles.

3. Find m∠SVW.

4. Find m∠VWT.

5. Vocabulary When two parallel lines are cut by a transversal,

angles are supplementary. (Explore Activity 1)

Find each angle measure.

5. m∠GDE =

6. m∠BEF =

7. m∠CDG =

YOUR TURN

6. What can you conclude about the interior angles formed when two

parallel lines are cut by a transversal?

CHECK-INESSENTIAL QUESTION?

Replace m∠VWZ with 3x° and m∠YVW with 6x°.

Combine like terms.


Simplify.

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Name Class Date

Vocabulary Use the figure for Exercises 7–10.

7. Name all pairs of corresponding angles.

8. Name both pairs of alternate exterior angles.

9. Name the relationship between ∠3 and ∠6.

10. Name the relationship between ∠4 and ∠6.

Find each angle measure.

11. m∠AGE when m∠FHD = 30°

12. m∠AGH when m∠CHF = 150°

13. m∠CHF when m∠BGE = 110°

14. m∠CHG when m∠HGA = 120°

15. m∠BGH =

16. m∠GHD =

17. The Cross Country Bike Trail follows a straight line

where it crosses 350th and 360th Streets. The two

streets are parallel to each other. What is the measure

of the larger angle formed at the intersection of the

bike trail and 360th Street? Explain.

18. Critical Thinking How many different angles would be formed by

a transversal intersecting three parallel lines? How many different angle

measures would there be?

Independent Practice21.1 8.EE.7b, 8.G.5

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19. Communicate Mathematical Ideas In the diagram at the

right, suppose m∠6 = 125°. Explain how to find the measures

of each of the other seven numbered angles.

20. Draw Conclusions In a diagram showing two parallel lines cut by a

transversal, the measures of two same-side interior angles are both given

as 3x°. Without writing and solving an equation, can you determine the

measures of both angles? Explain. Then write and solve an equation

to find the measures.

21. Make a Conjecture Draw two parallel lines and a transversal. Choose one

of the eight angles that are formed. How many of the other seven angles

are congruent to the angle you selected? How many of the other seven

angles are supplementary to your angle? Will your answer change if you

select a different angle?

22. Critique Reasoning In the diagram at the

right, ∠2, ∠3, ∠5, and ∠8 are all congruent,

and ∠1, ∠4, ∠6, and ∠7 are all congruent.

Aiden says that this is enough information

to conclude that the diagram shows two

parallel lines cut by a transversal. Is he

correct? Justify your answer.

FOCUS ON HIGHER ORDER THINKING

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EXPLORE ACTIVITY 1

ESSENTIAL QUESTION

Sum of the Angle Measures in a Triangle

What can you conclude about the measures of the angles of a triangle?

L ESSON

21.2Angle Theorems for Triangles

There is a special relationship between the measures of the interior

angles of a triangle.

Draw a triangle and cut it out. Label the angles A, B, and C.

Tear off each “corner” of the triangle. Each corner includes the

vertex of one angle of the triangle.

Arrange the vertices of the triangle around a point so that none

of your corners overlap and there are no gaps between them.

What do you notice about how the angles fit together around a point?

What is the measure of a straight angle?

Describe the relationship among the measures of the angles of △ABC.

The Triangle Sum Theorem states that for △ABC, m∠A + m∠B + m∠C = .

Reflect 1. Justify Reasoning Can a triangle have two right angles? Explain.

2. Analyze Relationships Describe the relationship between the two

acute angles in a right triangle. Explain your reasoning.

A

B

C

D

E

F

8.G.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Also 8.EE.7, 8.EE.7b

8.G.5

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EXPLORE ACTIVITY 2

Justifying the Triangle Sum TheoremYou can use your knowledge of parallel lines intersected by a transversal

to informally justify the Triangle Sum Theorem.

Follow the steps to informally prove the Triangle Sum Theorem. You should

draw each step on your own paper. The figures below are provided for you

to check your work.

Draw a triangle and label the angles as ∠1, ∠2, and ∠3 as

shown.

Draw line a through the base of the triangle.

The Parallel Postulate states that through a point not on

a line ℓ, there is exactly one line parallel to line ℓ. Draw

line b parallel to line a, through the vertex opposite the

base of the triangle.

Extend each of the non-base sides of the triangle to

form transversal s and transversal t. Transversals s and t

intersect parallel lines a and b.

Label the angles formed by line b and the transversals

as ∠4 and ∠5.

Because ∠4 and are alternate interior

angles, they are .

Label ∠4 with the number of the angle to which it is congruent.

Because ∠5 and are alternate interior angles,

they are .

Label ∠5 with the number of the angle to which it is congruent.

The three angles that lie along line b at the vertex of the triangle are

∠1, ∠4, and ∠5. Notice that these three angles lie along a line.

So, m∠1 + m∠2 + m∠5 = .

Because angles 2 and 4 are congruent and angles 3 and 5 are

congruent, you can substitute m∠2 for m∠4 and m∠3 for m∠5 in

the equation above.

So, m∠1 + m∠2 + m∠3 = .

This shows that the sum of the angle measures in a triangle is

always .

A

B

C

D

E

F

G

H

8.G.5

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Reflect 3. Analyze Relationships How can you use the fact that m∠4 + m∠1 +

m∠5 = 180° to show that m∠2 + m∠1 + m∠3 = 180°?

Finding Missing Angle Measures in TrianglesIf you know the measures of two angles in a triangle, you can use the Triangle

Sum Theorem to find the measure of the third angle.

Find the missing angle measure.

Write the Triangle Sum Theorem for

this triangle.

m∠D + m∠E + m∠F = 180°

Substitute the given angle measures.

55° + m∠E + 100° = 180°

Solve the equation for m∠E.

55° + m∠E + 100° = 180°

155°

-155°

__

+ m∠E = 180°

-155°

m∠E = 25°

So, m∠E = 25°.

EXAMPLEXAMPLE 1

STEP 1

STEP 2

STEP 3

Simplify.

Subtract 155° from both sides.

Find the missing angle measure.

YOUR TURN

4. 5.

m∠K = m∠R =

8.EE.7

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EXPLORE ACTIVITY 3

Exterior Angles and Remote Interior AnglesAn interior angle of a triangle is formed by two sides of the triangle. An exterior angle

is formed by one side of the triangle and the extension of an adjacent side. Each exterior

angle has two remote interior angles. A remote interior angle is an interior angle that is

not adjacent to the exterior angle.

• ∠1, ∠2, and ∠3 are interior angles.

• ∠4 is an exterior angle.

• ∠1 and ∠2 are remote interior angles to ∠4.

There is a special relationship between the measure of an

exterior angle and the measures of its remote interior angles.

Extend the base of the triangle and label the exterior angle

as ∠4.

The Triangle Sum Theorem states:

m∠1 + m∠2 + m∠3 = .

∠3 and ∠4 form a ,

so m∠3 + m∠4 = .

Use the equations in B and C to complete the following equation:

m∠1 + m∠2 + = + m∠4

Use properties of equality to simplify the equation in D :

The Exterior Angle Theorem states that the measure of an angle

is equal to the sum of its angles.

Reflect 6. Sketch a triangle and draw all of its exterior angles.

How many exterior angles does a triangle have

at each vertex?

7. How many total exterior angles does a triangle have?

A

B

C

D

E

8.G.5

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Math On the Spot

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Using the Exterior Angle TheoremYou can use the Exterior Angle Theorem to find the measures of the interior

angles of a triangle.

Find m∠A and m∠B.

Write the Exterior Angle

Theorem as it applies to this

triangle.

m∠A + m∠B = m∠ACD

Substitute the given angle measures.

(4y - 4)° + 3y° = 52°

Solve the equation for y.

(4y - 4)° + 3y° = 52°

4y° - 4° + 3y° = 52°

7y° - 4°

+ 4°

_

= 52°

+ 4°

_

7y° = 56°

7y° ___

7 = 56° ___

7

y = 8

Use the value of y to find m∠A and m∠B.

m∠A = 4y - 4

= 4(8) - 4

= 32 - 4

= 28

m∠B = 3y = 3(8)

= 24

So, m∠A = 28° and m∠B = 24°.

EXAMPLEXAMPLE 2

STEP 1

STEP 2

STEP 3

STEP 4

Remove parentheses.

Simplify.

Simplify.

Add 4° to both sides.

Simplify.


8. Find m∠M and m∠N.

m∠M =

m∠N =

YOUR TURN

Describe two ways to find m∠ACB.


8.EE.7b

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(18z + 3)°161°

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Guided Practice

Find each missing angle measure. (Explore Activity 1 and Example 1)

Use the Triangle Sum Theorem to find the measure of each angle

in degrees. (Explore Activity 2 and Example 1)

1.

m∠M =

2.

m∠Q =

3.

m∠T = , m∠U = ,

m∠V =

4.

m∠X = , m∠Y = ,

m∠Z =

Use the Exterior Angle Theorem to find the measure of each angle

in degrees. (Explore Activity 3 and Example 2)

7. Describe the relationships among the measures of the angles of a triangle.


5.

m∠C = , m∠D = ,

m∠DEC =

6.

m∠L = , m∠M = ,

m∠LKM =

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Name Class Date

Find the measure of each angle.

8.

m∠E =

m∠F =

9.

m∠T =

m∠V =

10.

m∠G =

m∠H =

m∠J =

11.

m∠Q =

m∠P =

m∠QRP =

12.

m∠ACB =

m∠BCD =

m∠DCE =

13.

m∠K =

m∠L =

m∠KML =

m∠LMN =

14. Multistep The second angle in a triangle is five times as large as the first.

The third angle is two-thirds as large as the first. Find the angle measures.

Independent Practice21.28.EE.7, 8.EE.7b, 8.G.5

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15. Analyze Relationships Can a triangle have two obtuse angles? Explain.

16. Critical Thinking Explain how you can use the Triangle Sum Theorem to

find the measures of the angles of an equilateral triangle.

17. a. Draw Conclusions Find the sum of the

measures of the angles in quadrilateral ABCD.

(Hint: Draw diagonal _

AC . How can you use the

figures you have formed to find the sum?)

Sum =

b. Make a Conjecture Write a “Quadrilateral Sum Theorem.” Explain

why you think it is true.

18. Communicate Mathematical Ideas Describe two ways that an exterior

angle of a triangle is related to one or more of the interior angles.


Unit 10680

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? ESSENTIAL QUESTIONHow can you determine when two triangles are similar?

Discovering Angle-Angle SimilaritySimilar figures have the same shape but may have different sizes. Two

triangles are similar if their corresponding angles are congruent and the

lengths of their corresponding sides are proportional.

Use your protractor and a straightedge to draw

a triangle. Make one angle measure 45° and

another angle measure 60°.

Compare your triangle to those drawn by your

classmates. How are the triangles the same?

How are they different?

Use the Triangle Sum Theorem to find the measure of the third angle

of your triangle.

Reflect1. If two angles in one triangle are congruent to two angles in another

triangle, what do you know about the third pair of angles?

2. Make a Conjecture Are two pairs of congruent angles enough

information to conclude that two triangles are similar? Explain.

A

B

C

EXPLORE ACTIVITY 1

L ESSON

21.3Angle-Angle Similarity

8.G.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Also 8.EE.6, 8.EE.7

8.G.5

681Lesson 21.3

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100° 100°

45°

35°

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70°

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8° 70°

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Math On the Spotmy.hrw.com

Using the AA Similarity Postulate

Explain whether the triangles are similar.

The figure shows only one pair of congruent angles. Find the measure of the

third angle in each triangle.

Because two angles in one triangle are congruent to two angles in the other

triangle, the triangles are similar.

EXAMPLE 1

45° + 100° + m∠C = 180° 100° + 35° + m∠D = 180°

145° + m∠C = 180° 135° + m∠D = 180°

145° + m∠C - 145° = 180° - 145° 135° + m∠D - 135° = 180° - 135°

m∠C = 35° m∠D = 45°

3. Explain whether the triangles are similar.

YOUR TURN

Are all right triangles similar? Why or

why not?


8.G.5

Angle-Angle (AA) Similarity Postulate

If two angles of one triangle are congruent to two angles

of another triangle, then the triangles are similar.

Unit 10682

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My Notes

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0.9 m

Net

Height of ballwhen hit

A B 12 m

h m

6 m

C

D

E

Finding Missing Measures in Similar TrianglesBecause corresponding angles are congruent and corresponding sides

are proportional in similar triangles, you can use similar triangles to solve

real-world problems.

While playing tennis, Matt is 12 meters from the net, which is 0.9 meter high.

He needs to hit the ball so that it just clears the net and lands 6 meters

beyond the base of the net. At what height should Matt hit the tennis ball?

In similar triangles, corresponding side lengths are proportional.

Matt should hit the ball at a height of 2.7 meters.

Reflect4. What If? Suppose you set up a proportion so that each ratio compares

parts of one triangle, as shown below.

Show that this proportion leads to the same value for h as in Example 2.

EXAMPLEXAMPLE 2

height of △ABC base of △ABC

height of △ADEbase of △ADE

BC ___ AB = DE ___ AD

AD ___ AB = DE ___ BC 6 + 12 ______

6 = h ___

0.9

0.9 × 18 ___ 6

= h ___ 0.9

× 0.9

0.9 × 3 = h

2.7 = h

8.EE.7

Substitute the lengths from the figure.

Use properties of equality to get h by itself.

Multiply.

Simplify.

Both triangles contain ∠A and a right angle, so △ABC and △ADE are similar.

683Lesson 21.3

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24 ft8 ft


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56 ft

Using Similar Triangles to Explain SlopeYou can use similar triangles to show that the slope of a line is constant.

Draw a line ℓ that is not a horizontal line. Label four points on the line

as A, B, C, and D.

You need to show that the slope between points A and B is the same as

the slope between points C and D.

A

5. Rosie is building a wheelchair ramp that is 24 feet long and 2 feet high.

She needs to install a vertical support piece 8 feet from the end of the

ramp. What is the length of the support piece in inches?

6. The lower cable meets the

tree at a height of 6 feet

and extends out 16 feet

from the base of the tree.

If the triangles are similar,

how tall is the tree?

YOUR TURN

EXPLORE ACTIVITY 2 8.EE.6

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Draw the rise and run for the slope between points A and B. Label the

intersection as point E. Draw the rise and run for the slope between

points C and D. Label the intersection as point F.

Write expressions for the slope between A and B and between C and D.

Slope between A and B: Slope between C and D:

Extend ‹ __

› AE and

‹ __

› CF across your drawing.

‹ __

› AE and

‹ __

› CF are both horizontal

lines, so they are parallel.

Line ℓ is a that intersects parallel lines.

Complete the following statements:

∠BAE and are corresponding angles and are .

∠BEA and are right angles and are .

By Angle–Angle Similarity, △ABE and are similar triangles.

Use the fact that the lengths of corresponding sides of similar

triangles are proportional to complete the following ratios: BE ___ DF = _____ CF

Recall that you can also write the proportion so

that the ratios compare parts of the same triangle: _____ AE = DF _____ .

The proportion you wrote in step shows that the ratios you wrote

in are equal. So, the slope of line ℓ is constant.

Reflect7. What If? Suppose that you label two other points on line ℓ as G and H.

Would the slope between these two points be different than the slope

you found in the Explore Activity? Explain.

B

C

BE _____ _____ CF

D

E

F

G

H

I H

C

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109°

41° 30°30°

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h ft

Mrs. Gilbert

Flagpole

16 ft 7.5 ft

5.5 ft

Guided Practice

1. Explain whether the triangles are similar. Label the angle measures in the

figure. (Explore Activity 1 and Example 1)

△ABC has angle measures and △DEF has angle

measures . Because in one

triangle are congruent to in the other triangle, the

triangles are .

2. A flagpole casts a shadow 23.5 feet long. At

the same time of day, Mrs. Gilbert, who is 5.5

feet tall, casts a shadow that is 7.5 feet long.

How tall in feet is the flagpole? Round your

answer to the nearest tenth. (Example 2)

3. Two transversals intersect two parallel lines as

shown. Explain whether △ABC and △DEC are

similar. (Example 1)

5.5 _____ = h _____

h = feet

∠BAC and ∠EDC are since they are .

∠ABC and ∠DEC are since they are .

By , △ABC and △DEC are .

4. How can you determine when two triangles are similar?


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4 ft

6 ft

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B

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85°

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47°

64°

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42°

5. Find the missing angle measures in the triangles.

6. Which triangles are similar?

7. Analyze Relationships Determine which angles are congruent to the

angles in △ABC.

8. Multistep A tree casts a shadow that is 20 feet long. Frank is 6 feet tall,

and while standing next to the tree he casts a shadow that is 4 feet long.

a. How tall is the tree?

b. How much taller is the tree than Frank?

9. Represent Real-World Problems Sheila is climbing on a ladder that

is attached against the side of a jungle gym wall. She is 5 feet off the

ground and 3 feet from the base of the ladder, which is 15 feet from the

wall. Draw a diagram to help you solve the problem. How high up the

wall is the top of the ladder?

10. Justify Reasoning Are two equilateral triangles always similar? Explain.

Use the diagrams for Exercises 5–7.

Name Class Date

Independent Practice21.38.EE.6, 8.EE.7, 8.G.5

687Lesson 21.3

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Work Area

h

6.5 cm 19.5 cm

3.4 cm

11. Critique Reasoning Ryan calculated the missing measure in the

diagram shown. What was his mistake?

12. Communicate Mathematical Ideas For a pair of triangular earrings, how

can you tell if they are similar? How can you tell if they are congruent?

13. Critical Thinking When does it make sense to use similar triangles to

measure the height and length of objects in real life?

14. Justify Reasoning Two right triangles on a coordinate plane are similar

but not congruent. Each of the legs of both triangles are extended by

1 unit, creating two new right triangles. Are the resulting triangles similar?

Explain using an example.

3.4 ___ 6.5

= h ____ 19.5

19.5 × 3.4 ___ 6.5

= h ____ 19.5

× 19.5

66.3 ____ 6.5

= h

10.2 cm = h


Unit 10688

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Ready

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106° 4y°

(3y + 22)°

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60 40

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MODULE QUIZ

21.1 Parallel Lines Cut by a TransversalIn the figure, line p∥line q. Find the measure of each angle if m∠8 = 115°.

1. m∠7 =

2. m∠6 =

3. m∠1 =

21.2 Angle Theorems for Triangles Find the measure of each angle.

4. m∠A =

5. m∠B =

6. m∠BCA =

21.3 Angle-Angle SimilarityTriangle FEG is similar to triangle IHJ. Find the missing values.

7. x = 8. y = 9. m∠H =

10. How can you use similar triangles to solve real-world problems?

ESSENTIAL QUESTION

689Module 21

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1 2

3

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A B D120°

2x°

x°

Assessment Readiness

MODULE 21 MIXED REVIEW

1. In the diagram, lines a and b are parallel and lines p and q are

parallel. Look at each pair of angles. Are the angles congruent?

Select Yes or No for A–C.

A. ∠1 and ∠2 Yes No

B. ∠3 and ∠4 Yes No

C. ∠5 and ∠6 Yes No

2. The variables x and y are related proportionally, and when x = 18, y = 12.

Choose True or False for each statement.

A. When x = 3, y = 5. True False

B. When x = 27, y = 18. True False

C. When x = 63, y = 42. True False

3. An artist is designing a triangular pane of glass

for a stained glass window, as shown in the

diagram. Given that ∠ABC is an exterior angle,

what is the measure of each interior angle of

the triangle? Explain your reasoning.

4. The diagram shows several of the boards that

form part of the frame of a roof. Is △JKN similar

to △JLM? Explain how you know.

690 Unit 10

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